In a set of three numbers, average of first two numbers is 2, average of last two numbers is 3, and average of first and last numbers is 4. What is average of three numbers?

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{three}}\,{\text{numbers}}\,{\text{be}}\,x,\,y,\,{\text{and}}\,z.\, \cr & {\text{We}}\,{\text{are}}\,{\text{given}}\,{\text{that}}\, \cr & \frac{{ {x + y} }}{2} = 2 \cr & \frac{{ {y + z} }}{2} = 3 \cr & \frac{{ {x + z} }}{2} = 4 \cr & {\text{Adding}}\,{\text{three}}\,{\text{equations}}, \cr & \frac{{ {x + y} }}{2} + \frac{{ {y + z} }}{2} + \frac{{ {x + z} }}{2} = 2 + 3 + 4 \cr & x + y + z = 9 \cr & {\text{The}}\,{\text{average}}\,{\text{of}}\,{\text{the}}\,{\text{three}}\,{\text{numbers}}\,{\text{is}}, \cr & \frac{{ {x + y + z} }}{3} = \frac{9}{3} = 3 \cr} $$